Journal of Function Spaces
Volume 2015 (2015), Article ID 767568, 8 pages
http://dx.doi.org/10.1155/2015/767568

Research Article

Formulae for the Generalized Drazin Inverse of a Block Matrix in Banach Algebras

Xiaoji Liu and Xiaolan Qin

Faculty of Science, Guangxi University for Nationalities, Nanning 530006, China

Received 15 June 2015; Accepted 1 October 2015

Academic Editor: Seppo Hassi

Copyright © 2015 Xiaoji Liu and Xiaolan Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present some new representations for the generalized Drazin inverse of a block matrix in a Banach algebra under conditions weaker than those used in resent papers on the subject.

1. Introduction

The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. Representations for the Drazin inverse of block matrices under certain conditions were given in the literature [1–5]. Generalized inverses of block matrices have important applications in automatics, probability, statistics, mathematical programming, numerical analysis, game theory, econometrics, control, and so on [6, 7]. In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a matrix in terms of its various blocks, where the blocks on the diagonal are required to be square matrices [7]. At the present time, there is no known complete solution to this problem.

In this paper, we present the representation for the Drazin inverse of under the conditions that and and that and (cf. Theorem 4). And we also give several representations for the Drazin inverse of under some weaker conditions.

Let be a complex Banach algebra with the unit. The sets of all Drazin invertible and quasinilpotent elements of will be denoted by and , respectively.

The generalized Drazin inverse of (introduced by Koliha in [8]) is the element which satisfies

If there exist the generalized Drazin inverse, then the generalized Drazin inverse of is unique and is denoted by .

Let and let be an idempotent (). We denote . Then we can write

Every idempotent induces a representation of an arbitrary element given by the following matrix:

Let and be the spectral idempotent of corresponding to . It is well known that can be represented in the following matrix form [9, Ch. 2]:

where , is invertible in the algebra , is its inverse in , and is quasinilpotent in the algebra . Thus, the generalized Drazin inverse of can be expressed as

Obviously, if , then is generalized Drazin invertible and .

2. Preliminary Results

First we start the following result which is proved in [10] for matrices, extended in [2] for a bounded linear operator and in [11] for arbitrary elements in a Banach algebra.

Lemma 1 (see [11, Theorem  2.3]). Let be a Banach algebra, , and an idempotent. Assume that and are represented as

(i)If and , then and are generalized Drazin invertible, and

 where

(ii)If and , then , and and are given by (7) and (8).

Lemma 2. Let , .

(1)[11, Corollary  3.4] If , then and

(2)[12, Theorem  2.2] If , then and

Lemma 3. Let be a Banach algebra and let have the representation (4), where . If and for some , then (in particular, if , then ). If and for some , then (in particular, if , then ).

Proof. From we get , , and . There exists such that . Since , we obtain . The proof for is similar.

3. Main Results

In [13, Theorem  3.2] authors gave an explicit representation for under conditions and . Here we replace the last condition by the condition ; we will get a much simpler expression for .

Theorem 4. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let . We can represent as in (4), where is invertible in the subalgebra and is quasinilpotent. Hence,

Let us write

From and

we get . Thus, and can be represented as

By Lemma 1 and , we get and

An elementary computation from (17) and leads to

Since ,

we get . An induction argument proves for any . Let us denote and observe that since , then is a subsequence of . Since is quasinilpotent, we get

Hence . By Lemma 2, we have that , because , , and . Also, Lemma 2 allows us to obtain

Again, from Lemma 1 and (16), we get that and

where

Observe that since , then . Hence, the expression of reduces to

From we get . Hence

So, we get

Observe that (22) and yield

We will express using (21) in terms only of and . We have from (17)

Hence from (21) we obtain

Also we have (we have written with an asterisk any entry whose exactly expression is not necessary)

From (27), (29), and (30), it follows (11). The inequality (12) trivially follows from (11).

If is a Banach algebra, then we can define another multiplication in by . It is trivial that is a Banach algebra. If we apply Theorem 4 to this new algebra, we can immediately establish the following result.

Theorem 5. Let be a Banach algebra and let . If there exist , , such that and , then and

Observe that we can obtain paired results as Theorems 4 and 5. We will not write explicitly these (trivially obtained) results.

Theorem 6. Let be a Banach algebra and let be such that . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . As in the proof of Theorem 4, from , we get .
From we have

Therefore, and . From Lemma 3 we get . Hence

Lemma 2 implies that because , , and . Also, we obtain

In this situation, we obtain

Since , we have . Thus (35) reduces to

From we have . Therefore,

so we get

We have

Also,

From (36), (38), (41), and (42), it follows (32).
The proof is finished.

As we have commented before, we can obtain a paired result by considering the Banach algebra with the product . The key hypothesis of this new result will be and .

Theorem 7. Let be a Banach algebra and let be such that . If there exist , , such that and , then and

Theorem 8. Let be a Banach algebra and let be such that . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . From , we have

The second equality of (45) implies for any . By Lemma 3 we get . Hence

Therefore, if denotes and denotes , then . Recall that is quasinilpotent, and thus . We have just proved or, equivalently, . Similarly we obtain .
Thus, and can be represented as

Since , by the proof of Theorem 4, we get . From Lemma 2, we have , because , , and . Also, Lemma 2 allows us to obtain

We have if and only if . But . If , then

The theorem is just proved.

Theorem 9. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . Since ,

we get and ; as in the proof of Theorem 4, we have . From Lemma 3 we get .
From we have

Therefore, , , and . From Lemma 3 we get and . Hence

Lemma 2 implies that because , , and ; as in the proof of Theorem 6, we obtain

In this situation, we obtain

Observe that since , then . Hence, the expression of reduces to

From and , we have

So we get

Hence

Thus (55) reduces to

We have

From (56), (62), and (63), it follows (50). The inequality (51) trivially follows from (50).
The proof is finished.

As we have commented before, we can obtain a paired result by considering the Banach algebra with the product . The key hypothesis of this new result will be and .

Theorem 10. Let be a Banach algebra and let . If there exist , , such that and , then and

In the rest of the paper, we will use some weaker conditions than in [11, Theorem  4.1]. For example, if we assume that and instead of and , we will get a much simpler expression for .

Theorem 11. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . We will use the condition : since

we get and ; as in the proof of Theorem 4, we have . From Lemma 3 we get .
From we have

Therefore, . Hence

where

The proof is finished.

Theorem 12. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . As in the proof of Theorem 11, from , we get and .
Since ,

we get . As in the proof of Theorem 11, we have

where

From , we get

The proof is finished.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11361009), the Guangxi Provincial Natural Science Foundation of China (no. 2013GXNSFAA019008), and Science Research Project 2013 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities. The authors would like to thank the referees and the editors for their suitable comments.

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